It doesn't really offer a lot. For a practical standpoint you have a choice of 15 cylinders (3 rows of 5), 21 cylinders (3 rows of 7) or 27 cylinders ( 3 rows of 9).
For the first adding an extra crank throw and the longer heavier crankcase and the more convoluted intake and exhaust systems needed for one extra cylinder rules it out pretty quick.
For the 2nd it gets a bit closer, 21 cylinders in 3 rows vs 18 in two. It is a 16.6% increase in displacement. Is that enough to pay for the longer crank and crankcase the more troublesome intake and exhaust?
Please note the Deerhound was in serious trouble in an open market. It started as a 2260 cu in engine with 7 overhead camshafts. It may have been in competition with the 2360 cu in Hercules and 2360 cu in Gnome-Rhone engines. With it's short stroke of 5in (127mm) it had the potential to run at higher rpm. However it would also be in competition with the Wright 14 cylinder R-2600.
The lined up cylinders present a bigger challenge to air cooling than staggered cylinders. You basically need the airflow to do two 90 degree turns, while the staggered cylinders require only gentile turns in the airflow to reach the cylinders and exit.
"A Deerhound II engine installed in an Armstrong Whitworth A.W.38 Whitley bomber. Note the relatively small diameter of the engine compared to that of the firewall. The 44 in (1.12 m) diameter Deerhound replaced the 51 in (1.29m) diameter Armstrong Siddeley Tiger that was originally installed in the Whitley II."
from "
Armstrong Siddeley 'Dog' Aircraft Engines"
However please note the area/volume of the overhead cam boxes and the area/volume NOT being used for fins/airflow to cool the cylinder heads (hottest part of the engine). upper and lower visible banks have cam covers removed.
Gnome Rhone engine
Note all push rods for valves come from 1 or 2 cam rings at the front of the engine. Trying to use pushrods from the front on a 3 rd row would be very difficult and adding cam ring/s at the back for a 3rd row increases cost/weight.
The three row arrangement is not as easy at it might appear.